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Properties

Label 2-1152-48.29-c2-0-28
Degree $2$
Conductor $1152$
Sign $-0.914 - 0.404i$
Analytic cond. $31.3897$
Root an. cond. $5.60265$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.900 − 0.900i)5-s − 7.66i·7-s + (−4.57 − 4.57i)11-s + (−10.9 − 10.9i)13-s + 0.0435i·17-s + (12.9 + 12.9i)19-s + 18.3·23-s − 23.3i·25-s + (−34.9 + 34.9i)29-s − 38.4·31-s + (−6.89 + 6.89i)35-s + (−11.3 + 11.3i)37-s + 45.6·41-s + (−51.4 + 51.4i)43-s + 56.5i·47-s + ⋯
L(s)  = 1  + (−0.180 − 0.180i)5-s − 1.09i·7-s + (−0.415 − 0.415i)11-s + (−0.844 − 0.844i)13-s + 0.00256i·17-s + (0.683 + 0.683i)19-s + 0.796·23-s − 0.935i·25-s + (−1.20 + 1.20i)29-s − 1.24·31-s + (−0.197 + 0.197i)35-s + (−0.307 + 0.307i)37-s + 1.11·41-s + (−1.19 + 1.19i)43-s + 1.20i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.914 - 0.404i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.914 - 0.404i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $-0.914 - 0.404i$
Analytic conductor: \(31.3897\)
Root analytic conductor: \(5.60265\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1),\ -0.914 - 0.404i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.2502451545\)
\(L(\frac12)\) \(\approx\) \(0.2502451545\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (0.900 + 0.900i)T + 25iT^{2} \)
7 \( 1 + 7.66iT - 49T^{2} \)
11 \( 1 + (4.57 + 4.57i)T + 121iT^{2} \)
13 \( 1 + (10.9 + 10.9i)T + 169iT^{2} \)
17 \( 1 - 0.0435iT - 289T^{2} \)
19 \( 1 + (-12.9 - 12.9i)T + 361iT^{2} \)
23 \( 1 - 18.3T + 529T^{2} \)
29 \( 1 + (34.9 - 34.9i)T - 841iT^{2} \)
31 \( 1 + 38.4T + 961T^{2} \)
37 \( 1 + (11.3 - 11.3i)T - 1.36e3iT^{2} \)
41 \( 1 - 45.6T + 1.68e3T^{2} \)
43 \( 1 + (51.4 - 51.4i)T - 1.84e3iT^{2} \)
47 \( 1 - 56.5iT - 2.20e3T^{2} \)
53 \( 1 + (44.6 + 44.6i)T + 2.80e3iT^{2} \)
59 \( 1 + (-20.7 - 20.7i)T + 3.48e3iT^{2} \)
61 \( 1 + (-2.42 - 2.42i)T + 3.72e3iT^{2} \)
67 \( 1 + (18.9 + 18.9i)T + 4.48e3iT^{2} \)
71 \( 1 - 114.T + 5.04e3T^{2} \)
73 \( 1 + 127. iT - 5.32e3T^{2} \)
79 \( 1 + 37.0T + 6.24e3T^{2} \)
83 \( 1 + (84.4 - 84.4i)T - 6.88e3iT^{2} \)
89 \( 1 + 136.T + 7.92e3T^{2} \)
97 \( 1 + 173.T + 9.40e3T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.253202456908331809071511935913, −8.084208851874246160965934544439, −7.60186283793552782602951202050, −6.79712432692399378639025497373, −5.59227957394124784760193898951, −4.87050090851028066649874587975, −3.76230054570457727635776768997, −2.90756877040565308201015026760, −1.31025908502519180895211484544, −0.07509696988083019458192341996, 1.89690366714613127617802358620, 2.74330911925414127747199985112, 3.94770285663433816542210637867, 5.13968691585496772803659208298, 5.63147947178810685245528101223, 7.00958516983774972120025815803, 7.37843640183449879462659123491, 8.567424633980865749115457902371, 9.330444220944281228425937698472, 9.795381799008332495348747583028

Graph of the $Z$-function along the critical line